Tatiana V. Bronnikova

Sub-harmonic resonance and multi-annual oscillations in northern mammals: a non-linear dynamical systems perspective

Abstract We conjecture that the well-known oscillations (3- to 5-yr and 10-yr cycles) of northern mammals are examples of subharmonic resonance which obtains when ecological oscillators (predator-prey interactions) are subject to periodic forcing by the annual march of the seasons. The implications of this hypothesis are examined through analysis of a bare-bones, Hamiltonian model which, despite its simplicity, nonetheless exhibits the principal dynamical features of more realistic schemes. Specifically, we describe the genesis and destruction of resonant oscillations in response to variation in the intrinsic time scales of predator and prey. Our analysis suggests that cycle period should scale allometrically with body size, a fact first commented upon in the empirical literature some years ago. Our calculations further suggest that the dynamics of cyclic species should be phase coherent, i.e., that the intervals between successive maxima in the corresponding time series should be more nearly constant than their amplitude—a prediction which is also consistent with observation. We conclude by observing that complex dynamics in more realistic models can often be continued back to Hamiltonian limits of the sort here considered.

Quasiperiodicity in a detailed model of the peroxidase–oxidase reaction

Quasiperiodicity in models of the peroxidase–oxidase reaction has previously been reported in ‘‘abstract’’ or phenomenological models which sacrifice chemical realism for tractability. In the present paper, we discuss how such behavior can arise in a detailed model (BFSO) of the reaction which has previously been shown to be consistent with experimental findings. We distinguish two types of quasiperiodic behavior. Regions of what we here refer to as ‘‘primary’’ quasiperiodicity are delimited by supercritical secondary Hopf bifurcations at one end of the relevant range of parameter values and by heteroclinic transitions at the other. Regions of so‐called ‘‘secondary quasiperiodicity’’ are delimited by supercritical Hopf bifurcations at both ends of the parameter range. The existence of a quasiperiodic route to chaos in a modified version of BFSO is also described. This paper emphasizes the experimental circumstances under which quasiperiodic dynamics may be detected in the lab and offers specific prescript...

Parametric dependence in model epidemics. I: Contact-related parameters

One of the interesting properties of nonlinear dynamical systems is that arbitrarily small changes in parameter values can induce qualitative changes in behavior. The changes are called bifurcations, and they are typically visualized by plotting asymptotic dynamics against a parameter. In some cases, the resulting bifurcation diagram is unique: irrespective of initial conditions, the same dynamical sequence obtains. In other cases, initial conditions do matter, and there are coexisting sequences. Here we study an epidemiological model in which multiple bifurcation sequences yield to a single sequence in response to varying a second parameter. We call this simplification the emergence of unique parametric dependence (UPD) and discuss how it relates to the model’s overall response to parameters. In so doing, we tie together a number of threads that have been developing since the mid-1980s. These include period-doubling; subharmonic resonance, attractor merging and subduction and the evolution of strange invariant sets. The present paper focuses on contact related parameters. A follow-up paper, to be published in this journal, will consider the effects of non-contact related parameters.

Secondary quasiperiodicity in the peroxidase–oxidase reaction

Secondary quasiperiodicity (period-doubled oscillations modulated by an incommensurate frequency), or “Q2”, is the temporal manifestation of quasiperiodic motion on period-doubled tori. The existence of this regime in a chemical reaction was first predicted (T. V. Bronnikova, W. M. Schaffer and L. F. Olsen, J. Chem. Phys., 1996, 105, 10849) in the course of numerical explorations of a detailed model of the peroxidase–oxidase system. Subsequent analysis (T. V. Bronnikova, W. M. Schaffer, M. J. B. Hauser and L. F. Olsen, J. Phys. Chem. B, 1998, 102, 632) suggested the possibility of homoclinic transitions (“fat torus” bifurcation) to chaos involving Q2. In the present paper, we present the first experimental evidence for secondary quasiperiodicity and fat torus bifurcations in a chemical oscillator. We also identify a second (“thin torus”) route to chaos involving Q2. The relationship of these two bifurcation scenarios to each other and to the experimental findings is discussed.

Controlling malaria: competition, seasonality and 'slingshotting' transgenic mosquitoes into natural populations.

Forty years after the World Health Organization abandoned its eradication campaign, malaria remains a public health problem of the first magnitude with worldwide infection rates on the order of 300 million souls. The present paper reviews potential control strategies from the viewpoint of mathematical epidemiology. Following MacDonald and others, we argue in Section 1 that the use of imagicides, i.e., killing, or at least repelling, adult mosquitoes, is inherently the most effective way of combating the pandemic. In Section 2, we model competition between wild-type (WT) and plasmodium-resistant, genetically modified (GM) mosquitoes. Under the assumptions of inherent cost and prevalence-dependant benefit to transgenics, GM introduction can never eradicate malaria save by stochastic extinction of WTs. Moreover, alternative interventions that reduce prevalence have the undesirable consequence of reducing the likelihood of successful GM introduction. Section 3 considers the possibility of using seasonal fluctuations in mosquito abundance and disease prevalence to ‘slingshot’ GM mosquitoes into natural populations. By introducing GM mosquitoes when natural populations are about to expand, one can ‘piggyback’ on the yearly cycle. Importantly, this effect is only significant when transgene cost is small, in which case the non-trivial equilibrium is a focus (damped oscillations), and piggybacking is amplified by the systems inherent tendency to oscillate. By way of contrast, when transgene cost is large, the equilibrium is a node and no such amplification is obtained.

Modeling Peroxidase-Oxidase Interactions

Reactive oxygen species (ROS) and peroxidase-oxidase (PO) reactions are Janus-faced contributors to cellular metabolism. At low concentrations, reactive oxygen species serve as signaling molecules; at high concentrations, as destroyers of proteins, lipids and DNA. Correspondingly, PO reactions are both sources and consumers of ROS. In the present paper, we study a well-tested model of the PO reaction based on horseradish peroxidase chemistry. Our principal predictions are these: 1. Under hypoxia, the PO reaction can emit pulses of hydrogen peroxide at apparently arbitrarily long intervals. 2. For a wide range of input rates, continuing infusions of ROS are transduced into bounded dynamics. 3. The response to ROS input is hysteretic. 4. With sufficient input, regulatory capacity is exceeded and hydrogen peroxide, but not superoxide, accumulates. These results are discussed with regard to the episodic nature of neurodevelopmental and neurodegenerative diseases that have been linked to oxidative stress and to downstream interactions that may result in positive feedback and pathology of increasing severity.Copyright

Parametric dependence in model epidemics. II: Non-contact rate-related parameters

In a previous paper, we discussed the bifurcation structure of SEIR equations subject to seasonality. There, the focus was on parameters that affect transmission: the mean contact rate, β0, and the magnitude of seasonality, ϵ B . Using numerical continuation and brute force simulation, we characterized a global pattern of parametric dependence in terms of subharmonic resonances and period-doublings of the annual cycle. In the present paper, we extend this analysis and consider the effects of varying non-contact-related parameters: periods of latency, infection and immunity, and rates of mortality and reproduction, which, following the usual practice, are assumed to be equal. The emergence of several new forms of dynamical complexity notwithstanding, the pattern previously reported is preserved. More precisely, the principal effect of varying non-contact related parameters is to displace bifurcation curves in the β0−ϵ B parameter plane and to expand or contract the regions of resonance and period-doubling they delimit. Implications of this observation with respect to modeling real-world epidemics are considered.